This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
3.1 Complete the table: Step 1: Analyze the pattern for "Number of matchsticks". The sequence is 3, 5, 7. The difference between consecutive terms is 5-3=2 and 7-5=2. This is an arithmetic sequence where each term increases by 2. Step 2: Complete the "Number of matchsticks" row. For Pattern 4: 7 + 2 = 9 For Pattern 10: Using the rule 1 + (2 × Pattern number), for Pattern 10, it's 1 + (2 × 10) = 1 + 20 = 21. Step 3: Complete the "Repeated addition" row. For Pattern 4: 1+2+2+2+2 For Pattern 10: 1 + (2 repeated 10 times) Step 4: Complete "The rule" row. For Pattern 4: 1+(2×4) For Pattern 10: 1+(2×10) Here is the completed table: | Pattern number | 1 | 2 | 3 | 4 | 10 | | :------------- | :-: | :-: | :-: | :-: | :-: | | Number of matchsticks | 3 | 5 | 7 | 9 | 21 | | Repeated addition | 1+2 | 1+2+2 | 1+2+2+2 | 1+2+2+2+2 | 1+(2 repeated 10 times) | | The rule | 1+(2×1) | 1+(2×2) | 1+(2×3) | 1+(2×4) | 1+(2×10) | 3.2 Draw the fourth stage in the space provided below: The pattern consists of triangles joined side-by-side. • Pattern 1: 1 triangle (3 matchsticks) • Pattern 2: 2 triangles (5 matchsticks) • Pattern 3: 3 triangles (7 matchsticks) • Pattern 4: 4 triangles (9 matchsticks) The fourth stage will be four triangles connected in a row, sharing vertical matchsticks. ` /\ /\ /\ /\ /__\/__\/__\/__\ ` (Please draw this in the provided box, ensuring the triangles are connected.) 3.3 Use the general rule to find the number of matchsticks you will use to build a pattern with 20 triangles. Step 1: Identify the general rule from the table. The general rule for the number of matchsticks (M) for a given pattern number (N) is: M = 1 + (2 × N) Step 2: Substitute the number of triangles into the rule. Here, the number of triangles is N = 20. M = 1 + (2 × 20) Step 3: Calculate the number of matchsticks. M = 1 + 40 M = 41 The number of matchsticks needed for 20 triangles is 41. Last free one today — make it count tomorrow, or type /upgrade for unlimited.